Topological properties of mechanical metamaterials

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NATALIA LERA VALVERDE

TESIS DOCTORAL MADRID 2019

TOPOL OGICAL PR OPER TI ES OF ME CH ANICAL MET AMA TERIALS NA TALIA LE RA

TOPOLOGICAL PROPERTIES OF MECHANICAL METAMATERIALS

UNIVERSIDAD AUTÓNOMA DE MADRID FACULTAD DE CIENCIAS

Topological properties of mechanical metamaterials Natalia Lera

Topological mechanics is a recent and intense area of research. It merges the knowledge of mathematical topology, electronic topology and topological defects with mechanical systems and phononic band structures.

Topology entered the field of condensed matter physics several decades ago through the study of topological defects of generic order parameters. The paradigmatic examples of these topological defects are solitons and vortices. However, it was the introduction of the concept of topological band theory that recently raised a major interest.

In this thesis, we strive to understand the basic ingredients for

non-trivial topology in mechanical systems arising from both

topological defects and band theory. As a result, new mechanical

models have been proposed and analyzed theoretically.

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Topological properties of mechanical metamaterials

This dissertation is presented by Natalia Lera

for the degree of

PhD in Condensed Matter Physics, Nanoscience and Biophysics Doctorado en F´ısica de la Materia Condensada, Nanociencia y Biof´ısica

Advised by

Prof. Jos´e Vicente ´Alvarez Carrera Departamento de F´ısica de la Materia Condensada

Universidad Aut´onoma de Madrid Madrid, March 2019

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Acknowledgments

First and foremost, I would like to thank my thesis advisor José Vicente Álvarez for the opportunity to pursue this degree and for all that he has taught me over these years. José Vicente, my gratitude for your encouragement and trust, for your careful explanations, being motivated all the time and providing insightful comments that help me develop the content in this thesis. I feel lucky for all these years working by your side.

I would also like to thank the people with whom I have collaborated during my PhD: Félix Yn- duráin, Jim Allen, Kai Sun, Daniel Torrent, Johan Christensen, Pablo San-José, Jinkyu Yang, Rajesh Chaunsali and Chun-Wei Chen from whom I have benefited greatly.

I thank José Soler, Julio Gómez, F.J. Vidal and Nicolás Agrait for their help with funding and ad- ministrative issues.

I would like to manifest my appreciation to the rest of the members in the department. Particularly, to my lunch group with whom I shared many laughs, thoughts and advice, Antonio, José, Hernán, Iván... Special mention to Miriam and Nacho who helped me integrate in the group, offered me a lot of delicious homemade food and whose help at the time of writing has been certainly important.

I would also like to thank Isidro Losada, Sahar Parkdel, Michelle Fritz and Simon Divilov with whom I shared the office in Madrid.

Special thanks to Krzysztof Bieniasz for carefully reading the manuscript and for the useful feedback received.

Outside the university, first I thank my parents, Araceli and Isidoro, and my sister, Ester, for their unconditional support in all my adventures.

For so many good conversations and long hours talking, walking and playing, I thank my old friends, Katheryna, Mar´ıa, Alberto and my new friends, Ely, Alex, Íñigo, Rubén, Gonzalo, Adrián, Jorge, Oscar, Alba, Almu, Shina, Silvia, Joaqu´ın, Manuel, Rakel, Natalia, Vanessa and others that come´ and go.

During my time in Ann Arbor, Jim Allen and Cathy Allen made my visit a delightful time, I appre- ciate the warming welcoming I received from them. My experience there was fantastic, thank you.

From Ann Arbor I have awesome stories, and even better memories; thanks Andira, Meredith, Paola, Carlos, Cristina, Vero and Manuel.

I would also like to thank my flatmates that over the years have made a great atmosphere in the house I call home, Sara, Vero, Carmen, Leoby, Natalia, Santos, Laura and Roc´ıo.

Thank you all, Natalia

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List of Figures

2.1 Mappings to the circumference space, S¹. . . 24

2.2 Mappings to the torus space, T². . . 24

2.3 Different configurations of x − y model. . . 25

2.4 Schematic representation of the SSH model. . . 29

2.5 Energy states and real space representation of the SSH model. . . 30

2.6 Chern insulator model. . . 33

2.7 Cartoon depicting Weyl Hamiltonians in two and three dimensions. . . 35

2.8 Weyl semimetal band structure and vortex in the phase of eigenvectors. . . 35

2.9 Maxwell lattice in two dimensions. . . 38

2.10 Plate crystal. . . 40

3.1 Maxwell lattices that satisfy the Maxwell rule, one is isostatic and the other is not. . . 44

3.2 Topological one dimensional Maxwell lattice equivalent to SSH chain. . . 47

3.3 Band structure of the insulating and topological one dimensional chain. . . 48

3.4 Square Maxwell lattice and band energy. . . 49

3.5 Kagome Maxwell lattice and enumeration of the unit cell. . . 50

3.6 Determinant of C(k) in the complex space. . . 54

3.7 Zero modes of the one dimensional chain. . . 54

3.8 Poles of the inverse dynamical matrix D⁻¹(k) in the complex k-space. . . 57

3.9 Directional response for the topological one dimensional chain. . . 57

4.1 Quantum dot model in BDI topological class, energy levels and topological invariant. . 63

4.2 Winding of determinant ˜C(k) for the quantum dot model. . . 63

4.3 Mechanical model in d = 0, energy levels and topological invariant. . . 65

4.4 Schematic representation of the singular eigenvectors of C at the two sides of a topological transition. . . 67

4.5 Representation of zero eigenvalues of HC. One zero mode and one state of self stress. . 67

5.1 Perimeter constraint. . . 71

5.2 Rotors used as a single degree of freedom in the unit cell. . . 72

5.3 1D model with triangular perimeter constraint. . . 73

5.4 Gap structure of the 1D model in parameter space and topological phase diagram. . . 74

5.5 Zero modes arising in finite 1D model. . . 74

5.6 2D model with triangular perimeter constraint, gap value in parameter space and topological index in x and y directions. . . 76

5.7 Cut of the gap value and topological invariant for fixed parameter. Spectrum for gapless system and vortices in the determinant phase due to Weyl points. . . 76

5.8 3D model with triangular perimeter constraint. . . 77

5.9 Gap value of the 3D model, topological invariant phase diagram and Weyl lines. . . . 78 9

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10 LIST OF FIGURES 5.10 Cut of the gap and topological invariant for fixed r₁ = r₂ = r₃ = 0.9 in the 3D model

and Weyl lines appearing at gapless models. . . 78

5.11 2D model with quadrangular perimeter constraint, gap value and topological invariant. 79 5.12 Directional response for several phases of the 1D system. . . 80

5.13 Directional response of several phases of the 2D system. . . 81

6.1 Bands of the square lattice plate crystal, method 1 testing accuracy. . . 89

6.2 Bands of the square lattice plate crystal, comparison of the two methods. . . 89

6.3 First Bragg gap in the square lattice plate crystal. . . 90

6.4 Band structure of honeycomb plate crystal . . . 90

6.5 Berry curvature for the honeycomb lattice. . . 93

6.6 Valley Chern number in honeycomb lattice. . . 93

6.7 Ribbon with square lattice arrangement. . . 94

6.8 Band structure of the square lattice ribbon and mode shape. . . 95

6.9 Ribbon with supercell for honeycomb lattice, band structure and midgap states. . . . 96

6.10 Deformation field solution |w| for a single resonator. . . 98

6.11 Deformation field |w| for a finite cluster of resonators with square arrangement. . . 99

6.12 Z-shape interface in honeycomb lattice. . . 99

6.13 Deformation field without external force. . . 100

6.14 Behavior of the minimum value of the singular value decomposition for the natural modes as a function of the inverse system size. . . 100

7.1 Undistorted kagome lattice, deformation parameters and Brillouin Zone. . . 103

7.2 Dirac frequency as a function of plate crystal parameters and band structure. . . 103

7.3 Deformed kagome lattices. . . 104

7.4 Gap closings in parameter space at K in kagome lattice. . . 104

7.5 Spring-mass model of kagome lattice . . . 107

7.6 Spatial symmetric and non-symmetric kagome lattice band structure. . . 107

7.7 Energy level at K point and M operator representation. . . 107

7.8 Band structure of kagome plate crystal. . . 108

7.9 Berry curvature for the two valley phases. . . 109

7.10 Mode shapes at K point. . . 109

7.11 Valley Chern number as a function of f . . . 110

7.12 Ribbon of resonators with an interface over an infinite plate, band structure and topological mode shapes. . . 110

7.13 Ribbon of resonators with an interface of different type over an infinite plate, band structure and topological mode shapes. . . 111

7.14 Ribbon of resonators with an armchair type of interface over an infinite plate, band structure and topological mode shapes. . . 112

7.15 Absolute value of the plate displacement field of the topological edge state. . . 113

7.16 Absolute value of the plate displacement field of the topological edge state with disorder.113 7.17 Plate displacement field Re(w) for multipoint dephased excitation. One-sided edge modes.114 7.18 MST simulations of a square waveguide with and without disorder. . . 114

7.19 Distorted kagome lattice, spring-mass model. . . 116

7.20 Energy levels at K point for distorted kagome model. . . 116

7.21 Band structure of deformed kagome lattice for several α-values. . . 117

7.22 Berry curvature. . . 117

7.23 Valley Chern number as a function of α. . . 118

7.24 Ribbon of resonators with a topological interface over an infinite plate, band structure and mode shape. . . 119

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LIST OF FIGURES 11 7.25 Ribbon of resonators with a topological interface over an infinite plate. The interface

mixes valleys. Band structure and mode shape. . . 120

7.26 Plate displacement field Re(w) for multipoint dephased excitation. One-sided edge modes.120 8.1 Kekul´e supercell. . . 124

8.2 Trivial Kekul´e distortion. . . 124

8.3 Brillouin Zone and band structure of honeycomb lattice. Undistorted Kekul´e. . . 125

8.4 Band structure for trivial Kekul´e distortion. . . 125

8.5 Gap structure in parameter space for trivial Kekul´e distortion. . . 125

8.6 Density of states for flexural waves in trivial Kekulé distorted lattice for fixed parameters.126 8.7 Density of states for flexural waves in trivial Kekulé distorted lattice in parameter space.126 8.8 Non trivial Kekulé arrangement of resonators. . . 128

8.9 Kekul´e vortex in real space. . . 129

8.10 Local density of states for trivial and non-trivial Kekul´e. . . 129

8.11 Frequency peaks in the gap for non-trivial distorted Kekul´e. . . 130

8.12 Mode shape of the three peaks appearing in LDOS. . . 130

8.13 MST simulations for non-trivial Kekul´e. . . 131

8.14 Local density of states for non-trivial Kekul´e with invariant n = 2. . . 132

8.15 MST simulations for non-trivial Kekul´e with invariant n = 2. . . 132

8.16 Frequency drift of the peaks for mass loaded resonators at the core in non-trivial Kekul´e134 8.17 Band structure for the periodic system with undistorted pattern and different set of plate parameters. . . 134

8.18 Frequency drift of non-topological peaks for appropriate combination of resonator mass and frequency change at the core in non-trivial Kekul´e . . . 135

8.19 MST simulation of the topological mode with Gaussian disorder in non-trivial Kekul´e. 136 8.20 MST simulations and LDOS of the topological mode in non-trivial Kekul´e after random removal of resonators. . . 136

8.21 MST simulations and LDOS of the topological mode in non-trivial Kekul´e after removal of resonators inside a given circle. . . 137

8.22 MST simulations and frequency of the topological mode in non-trivial Kekul´e after removal of resonators at the core. . . 138

A.1 Plate differential element of thickness h with all moments and forces acting on the differential element and Bessel functions. . . 144

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List of symbols and abbreviations

π_n nth-homotopy group Sⁿ n-sphere

∼= isomorphic to ν winding number

k momentum

u_k eigenvector of momentum k R set of real numbers

Z set of integers

C set of complex numbers

θ angular coordinate in polar coordinates Q(k) projector operator at momentum k

q(k) off-diagonal part of the projector operator at momentum k

~e_x unitary vector in x direction. Equivalent definition for y or z directions.

κ spring constant dof degree of freedom UC unit cell

BZ Brillouin Zone

PHS Particle-hole symmetry TRS Time reversal symmetry Part I: Maxwell lattices

M mass

D dynamical matrix C compatibility matrix Q equilibrium matrix

u_i displacement of coordinate i em elongation of bond m f_i force along coordinate i t_m tension of bond m ZM zero modes

SSS state of self-stress

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14 CHAPTER 0. LIST OF SYMBOLS AND ABBREVIATIONS Part II: Plate crystals

w displacement field

D plate stiffness, also Dynamical matrix ρ plate mass density

Ω_R dimensionless resonator frequency Ω_D dimensionless Dirac frequency γ dimensionless resonator mass C_v valley Chern number

f , α deformation parameters in kagome lattices MST multiple scattering theory

C₃ 3-fold rotational symmetry

K and K⁰ Momenta at the corners of the hexagonal Brillouin Zone

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Chapter 1 Introduction

The purpose of this thesis is to understand the mechanisms of topological mechanics and design mechanical metamaterials supporting topological bound modes.

Topological mechanics is a recent and intense area of research. It merges the knowledge of mathematical topology, electronic topology and topological defects with mechanical systems and phononic band structures. Here, we will focus on time reversal symmetric mechanical structures.

In this introduction we will present the state of the art research in the field, specify the goals of this work, establish the methodology and, finally, we will describe the structure of the thesis.

Topology entered the field of condensed matter physics several decades ago through the study of topological defects of generic order parameters, either crystalline, magnetic, or superconducting. The paradigmatic examples of these topological defects are solitons and vortices. However, it was the introduction of the concept of topological band theory that recently raised a major interest. The insulating Hamiltonians of periodic lattices in d spatial dimensions can be understood as a map from a d-loop Brillouin Zone to the space of Hamiltonians. Whether these mappings can be transformed continuously from one another is determined by the d-th homotopy group. The classification table of electronic topological insulators and superconductors tells us the type of topological invariant char- acterizing any given system based on its spatial dimension d and three symmetries: time-reversal symmetry, particle-hole symmetry and chiral symmetry. A Hamiltonian with a topological invariant homotopically different from the vacuum is characterized by edge states in the finite system. This means that there is no continuous connection between the system bulk and the vacuum. Therefore, the existence of edge states in topological insulators is protected and cannot be removed. This is the bulk-edge correspondence principle. This phenomenon is related to the binding of topological states to a vortex core, which also supports a homotopy group description. In the latter case, the homotopy group description depends on the dimension of the defect and the space where it is embedded.

Mechanical metamaterials are fabricated structures with novel properties arising from their macroscopic design, not their microscopic crystalline structure. Many structures with distinctive behavior such as negative Poisson ratio, zero elastic moduli or negative compressibilities were designed. How- ever, the field was revolutionized with the introduction of topology. The advantageous idea of having protected states robust against disorder together with the accessible design of macroscopic structures, led to a rapid development of the field. The major interest in topological mechanical metamaterials came from the new ways of conveying energy through topological waveguides or binding wave states to topological defects. Another field of application is soft matter physics and biology where topological modes could play a role in rigidity transitions or jamming in mechanical structures and systems like biological tissue or foams. Metamaterials are a perfect playground to test and study these properties.

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16 CHAPTER 1. INTRODUCTION From the fundamental point of view, topological mechanical metamaterials support the two types of topology relevant for condensed matter physics that we have mentioned above. On the one hand, phononic band topology follows the same mapping from a d-loop Brillouin Zone to the dynamical matrix (or its square root). On the other hand, topological defects can be implemented with mechanical degrees of freedom as well.

Although examples of time-reversal symmetry breaking systems exists, such as systems with dissipa- tion or gyroscopic magnets, in this thesis we will focus on time reversal symmetric systems.

Our primary objective is to create and characterize different metamaterials that support a topological description. First, we aim to design a topological Maxwell lattice. Especially interesting are the insulating phases in three dimensions and Weyl phases in two and three dimensions. There were few previous examples and they contained a large number of degrees of freedom, which limited the possibility of analytic description. As a second goal, we seek to show examples of the zero topological phase with Maxwell lattices. In order to implement this goal, we will create a model supporting this kind of topology and then study its implications. The second part intends to extend the valley Hall effect to more complex unit cells than graphene and study the new arising phenomena. In the last stage of this work, we aim to understand the behavior of mechanical systems with a vortex in their crystal order.

In this thesis we strive to understand the basic ingredients for non-trivial topology in electronic systems. Then, we consider the nuances of mechanical systems. Next, we design simple unit cells with appropriate connectivity and symmetries which are different from the already existing designs. Fi- nally, we characterize the arising topological modes and test their robustness.

The methodology we have developed and used relies on a broad variety of mathematical and compu- tational tools. Mostly, the methodology aims to solve band structures and density of states, computing energy levels in finite lattices, local density of states and topological invariants.

In the first part, we analytically compute compatibility matrices in zero, one, two and three spatial dimensions for periodic and open boundary conditions or a combination of the two.

In the second part, we solve the wave equation for the flexural dynamics on a thin plate. For periodic systems we use methods based on the Bloch Theorem, i.e. Plane Wave Expansion (PWE) which provide the band structure, densities of states and spatial struture of normal modes. For plates coupled to clusters of resonators or for disordered sytems, we use Multiple Scattering Theory (MST) and compute the plate displacements and local densities of states. For semi-periodic systems we apply a combination of both methods.

Concerning topological invariants, we use integration methods of products of eigenvectors. In one dimension we control the phase of eigenvectors, in two dimensions we use an algorithm introduced by Fukui, Hatsugai and Suzuki without gauge-fixing conditions of eigenvectors.

Regarding the structure of the thesis, it is divided in two parts. In the first part, we consider Maxwell lattices, whose topology is equivalent to the BDI class of a canonical classification of electronic topology. Thanks to a clever and unique decomposition of the dynamical matrix, we can define a topological invariant. This topological invariant indicates whether the topological zero modes are localized on the left or on the right edge. In this thesis, we extend the topological invariant to zero- dimensional Maxwell lattices. We also describe a new mechanical constraint compatible with Maxwell lattice theory. This new constraint allows for the construction of topological materials with higher value of the topological invariant. This means that there are several edge states that can be localized at different edges in different combinations. We manage to construct topological insulating models in one, two and three dimensions. With this new constraint we find regions in the parameter space where the system is a Weyl semimetal.

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17 In the second part of this thesis, we study flexural waves in plate crystals. We create and study topological edge states appearing at the boundary between two different Hall phases in kagome lattices.

Varying the parameters of the deformation we find two valley Hall transitions, one breaking spatial inversion symmetry (as the case of the already studied honeycomb lattice) or the other breaking mirror symmetry. We manage to create one-sided edge states which are relevant in waveguiding applications.

We also create a vortex distortion of the honeycomb lattice following a Kekul´e pattern. We can bind wave modes to the vortex. The excitation of these modes do not propagate, they are localized at the vortex core and they are robust against disorder.

This thesis is organized as follows, chapter 2 is common to the two parts of the thesis. We introduce basic mathematical concepts of topology, topological defects and electronic topology together with tools to compute topological invariants and why they appear. In the last part of this chapter, we explain why mechanical systems have topology and introduce briefly Maxwell lattices and plate crystals.

Thereafter, the text is divided in two parts. Part I considers Maxwell lattices.

In chapter 3, we discuss the concepts and methods to deal with Maxwell lattices and its topological properties.

In chapter 4, results concerning zero topology are presented for BDI systems. We present a topological Maxwell lattice in which we can tune the topological invariant and explain the meaning of each phase.

In chapter 5, we present and discuss results for three different mechanical models designed in one, two and three dimensions. Each model has a few parameters to adjust three possibilities for the topological invariant value. This design opens the door for higher value of the topological invariant.

In part II, we move on to plate crystals.

In chapter 6, methodology of plate crystals is introduced together with examples in square lattice and honeycomb lattice.

In chapter 7, we study kagome lattices. We parameterize a deformation of the lattice with several symmetries. Varying the parameters of the lattice deformation we find gap openings and closings that signal different topological transitions. Either by breaking inversion symmetry or mirror symmetry in two different configurations, we end up with phases analogous to quantum valley Hall effect.

In chapter 8, a distorted honeycomb lattice with Kekul´e pattern is studied. We find a topological bound state, we prove its protection and robustness.

In chapter 9, the main conclusions about the results presented in this thesis and ideas for future work are discussed.

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Introducci´ on

El objetivo de esta tesis es entender los mecanismos de la topolog´ıa mecánica y diseñar metamateriales mecánicos capaces de albergar modos topológicos ligados.

La topolog´ıa mecánica es un campo de investigación nuevo y con intensa actividad. Combina el conoci- miento de la topolog´ıa matemática, topolog´ıa electrónica y defectos topológicos con sistemas mecánicos y estructuras de bandas fonónicas. En esta tesis nos centraremos en estructuras mecánicas con simetr´ıa temporal.

En esta introducci´on presentaremos los ´ultimos avances en el campo, especificaremos los obejetivos de este trabajo y estableceremos la metodolog´ıa. Finalmente, describiremos la estructura de esta tesis.

La topolog´ıa está presente en la f´ısica de la materia condensada desde hace varias décadas debido al estudio de defectos topológicos en parámetros de orden como por ejemplo, en el orden cristalino, magnético o superconductor. Ejemplos clásicos de defectos topológicos son solitones o vórtices. Sin embargo, fue la topolog´ıa de bandas lo que ha acaparado un mayor interés en los últimos años. Los Hamiltonianos aislantes de redes periódicas en d dimensiones se pueden entender como una correspondencia uno a uno entre la zona de Brillouin, una curva cerrada en d dimensiones, y el espacio de Hamiltonianos. El n-ésimo grupo de homotop´ıa informa si estas correspondencias (mapas) pueden transformarse unas en otras de forma continua o no. La classificación de aislantes y superconductores topológicos clasifica el tipo de invariante de un Hamiltoniao en función de su dimensión espacial d y de tres simetr´ıas: simetr´ıa temporal, simetr´ıa part´ıcula-hueco y simetr´ıa quiral. Un Hamiltoniano con un invariante topológico homotopicamente diferente del vac´ıo está caracterizado por estados de borde en los sistemas finitos. Esto significa que no hay una correspondencia continua entre el sistema periódico y el vac´ıo, lo que implica la existencia de estados de borde en el aislante que están protegidos y no se pueden eliminar. Este es el principio de correspondencia entre el sistema periódico y el borde. Los estados ligados a un defecto topológico también admiten una descripción en términos de grupos de homotop´ıa. En este último caso, el grupo de homotop´ıa depende de las dimensiones tanto del defecto como del espacio en el que se encuentra.

Los metamateriales mec´anicos son estructuras fabricadas con propiedades diferentes a la naturaleza.

Estas propiedades surgen de su diseño macroscópico y no de su estructura cristalina. Muchas estructuras con propiedades diferentes han sido diseñadas, por ejemplo, materiales con coeficiente de Poisson negativo, módulo elástico cero o compresibilidades negativas. Sin embargo, el campo sufrió una revo- lución con la introducción de la topolog´ıa. La ventaja de tener estados protegidos y robustos frente al desorden junto con un diseño fácil de implementar, hizo que el campo se desarrollara muy rápido en los últimos años. El mayor interés en topolog´ıa de metamateriales mecáncios proviene de las nuevas formas de conducir la energ´ıa a través de gu´ıas de onda o atrapando ondas en defectos topológicos.

Otro campo de aplicación es la f´ısica de la materia blanda donde los modos topológicos podr´ıan jugar un papel en transiciones de rigidez en estructuras mecánicas y en sistemas como tejidos biológicos o espumas. Los metamateriales son sistemas perfectos para probar y estudiar estas propiedades.

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20 CAPÍTULO 1. INTRODUCCI ÓN Desde el punto de vista fundamental, la topolog´ıa mecánica de metamateriales puede tener los dos tipos de topolog´ıas relevantes en f´ısica de la materia condensada que hemos mencionado. Por un lado, la topolog´ıa de bandas fonónicas sigue un patrón similar y podemos definir grupos de homotop´ıa desde una zona de Brillouin hasta una matriz dinámica (o su ra´ız cuadrada). Por otro lado, los defectos topológicos pueden definirse también con grados de libertad mecánicos.

Aunque existen ejemplos de sistemas que rompen la simetr´ıa temporal, por ejemplo sistemas con disi- pación o con giróscopos magnéticos, en esta tesis nos centraremos en sistemas con simetr´ıa temporal.

Nuestros objetivos son la creación y caracterización de sistemas diferentes con propiedades topológi- cas no triviales. También caracterizar esos modos. Primero nos propusimos crear una red de Maxwell con topolog´ıa no trivial. Especialmente quer´ıamos diseñar aislantes en tres dimensiones y fases de Weyl tanto en dos como en tres dimensiones ya que hab´ıa pocos ejemplos. Como segundo objetivo quer´ıamos hallar una red de Maxwell en cero dimensiones y estudiar su significado. En el campo de los metamateriales en placas delgadas nuestro primer objetivo fue extender el efecto Hall de valle a sistemas con celda unidad más compleja que el grafeno y estudiar toda la fenomenolog´ıa nueva aso- ciada. Por último, comprobar como se comportan los sistemas mecánicos con un vórtice en su orden cristalino.

Primero entenderemos los ingredientes básicos de la topolog´ıa en sistemas electrónicos. Después con- sideraremos las peculiaridades de los sistemas mecánicos. Por ejemplo, la simetr´ıa part´ıcula-hueco no es exacta, o la facilidad de estos sistemas de sintonizar la frequencia de excitación (ser´ıa análogo a cambiar el nivel de Fermi libremente). Luego, diseñaremos celdas unidad con la connectividad necesaria y simetr´ıas que sean diferentes de los modelos ya existentes.

La metodológia que hemos desarrollado y empleado consiste en un amplio abanico de herramientas matemáticas y computacionales. Especialmente, la metodolog´ıa pretende resolver estructuras de bandas, calcular densidades de estados, niveles de eneg´ıa en sistemas finitos, densidades locales de estados e invariantes topológicos.

En la primera parte, calculamos matrices de compatibilidad de forma anal´ıtica en todas las dimensiones espaciales (de cero a tres) para sistemas periódicos, abiertos o combinaciones de ambos. En la segunda parte, resolvemos la ecuación de láminas delgadas. Para sistemas periódicos, utilizamos méto- dos basados en el Teorema de Bloch, es decir, método de expansión en ondas planas (PWE) que nos dará la estructura de bandas, densidades de estados y la estructura espacial de los modos normales.

Para láminas acopladas a un grupo de resonadores o para sistemas desordenados, emplearemos teor´ıa de scattering múltiple (MST). Calcularemos los desplazamientos de la placa y la densidad local de estados. Para sistemas semi-periódicos aplicaremos una extensión del primer método.

Respecto a los invariantes topológicos, utilizamos métodos de integración de producto de autovectores.

En una dimensi´on controlamos la fase de los autovectores, en dos dimensiones utilizamos un algoritmo dise˜nado por Fukui, Hatsugai y Suzuki que no requiere fijar el gauge de los autovaleres.

La estructura de este trabajo est´a dividida en dos partes. En la primera parte, estudiamos redes de Maxwell cuya topolog´ıa es equivalente a la clase BDI en sistemas electr´onicos. Gracias a una ingeniosa y

´

unica descomposición de la matriz dinámica, podemos definir un invariante topológico. Este invariante contiene la información necesaria para determinar si un modo de cero energ´ıa estará localizado a la derecha o izquierda del sistema abierto. En esta tesis extendemos la noción de invariante topológico en redes de Maxwell a sistemas con cero dimensiones espaciales. También describimos una nueva ligadura mecánica compatible con la teor´ıa de redes de Maxwell. Esta nueva ligadura permite la construcción de materiales topológicos con un invariante topológico mayor. Esto significa que hay varias fases y varios modos de borde que pueden estar localizados en diferentes combinaciones a derecha o izquierda.

Hemos conseguido construir modelos de aislantes topol´ogicos en una, dos y tres dimensiones espaciales.

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21 Con la nueva ligadura encontramos regiones del espacio de parámetros donde existen fases topológicas semimetálicas de Weyl.

En la segunda parte de esta tesis estudiamos ondas flexurales de láminas delgadas. Creamos y carac- terizamos modos topológicos de borde que aparecen en el borde entre dos fases distintas de Hall de valle en la red kagome. Diseñamos dos tipos de transiciones de valle. Primero, rompiendo la simetr´ıa espacial de inversión (como en el caso ya estudiado para la red hexagonal) y una segunda, rompiendo una simetr´ıa de reflexión. Además, conseguimos crear modos unilaterales que son relevantes para aplicaciones de gu´ıas de onda. También creamos un vórtice en la red hexagonal siguiendo un patrón Kekulé. Este sistema es capaz de localizar una onda. La excitación no se propraga y su excitación es robusta ante desorden.

Esta tesis está organizada como se especifica a continuación. El cap´ıtulo 2 es común a ambas partes de la tesis. Es una introducción a los fundamentos de la topolog´ıa matemática, topolog´ıa de defectos y a la topolog´ıa electrónica. También se incluyen herramientas para calcular invariantes topológicos y por qué son distintos de cero. En la última parte del cap´ıtulo, explicaremos por qué los sistemas mecánicos también tienen topolog´ıa e introduciremos brevemente las redes de Maxwell y los cristales en láminas delgadas.

De aqu´ı en adelante, la tesis est´a dividida en dos partes. La primera parte considera las redes de Maxwell.

En el cap´ıtulo 3, se discuten conceptos y m´etodos para tratar a las redes de Maxwell y sus propiedades topol´ogicas.

En el cap´ıtulo 4, se presentan resultados de topolog´ıa en cero dimensiones para sistemas en clase BDI.

Presentamos una red de Maxwell topol´ogica en la que podemos cambiar el invariante topol´ogico. Se explica el significado de cada fase.

En el cap´ıtulo 5, presentamos y discutimos los resultados para tres modelos mecánicos diseñados en una, dos y tres dimensiones. Cada uno de los tres modelos tiene diferentes invariantes y mostraremos como una ligadura nueva abre la puerta para invariantes topológicos más altos. Diseñamos además un modelo en una dimensión con cuatro fases topológicas distintas.

En la parte II, pasamos a cristales sobre l´aminas delgadas.

En el cap´ıtulo 6, se introduce la metodolog´ıa para tratar l´aminas delgadas junto con ejemplos en red cuadrada y hexagonal.

En el cap´ıtulo 7, estudiamos la red de kagome. Parametrizaremos una deformación de la red con varias simetr´ıas. Un cierre y reapertura del gap caracterizará dos tipos de transición topológica. Ya sea rompiendo la simetr´ıa de inversión o la de reflexión en dos configuraciones diferentes, tendremos fases análogas al efecto cuántico Hall de valle.

En cap´ıtulo 8, se estudia una distorsión de la red hexagonal con patrón Kekulé. Hallamos un modo topológico y probamos su protección y robustez.

En el cap´ıtulo 9, se presentan las conclusiones principales de esta tesis as´ı como una discusi´on de los retos para futuras investigaciones.

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Chapter 2 Fundamentals of Topology and Mechanical Metamaterials

Topological mechanics combines the knowledge in mathematical and electronic topology with elastic equations. It has become an intensive area of research in the last few years. A proof of that is March Meeting 2019 hosts a total of five sesions in mechanical metamaterials and topological mechanics. In this introduction, we first describe the basic mathematical concepts that lead to the understanding of topological defects and electronic topology. Later, we continue with electronic topology concepts established in condensed matter physics, such as topological invariants, the integer quantum Hall effect (IQHE), the symmetry classes in Hamiltonians and the classification table. Special attention will be given to BDI class in one spatial dimension and the Chern insulator in two dimensions. Concerning topological mechanics, we describe Maxwell lattices and its zero frequency topology. Finally, we use plate dynamics as a two dimensional system to study how topological defects modify the behavior of flexural waves.

2.1 Mathematical concepts

Donuts and cups are topologically equivalent. However, they are not equivalent to a sphere. To put this in mathematical language, we introduce the concept of

Homotopy: two continuous functions are homotopic (from Greek, homos “similar”, topos “place”) if one can be continuously deformed into the other. The fundamental group studies the homotopy of one variable functions. We define a loop as a mapping from I = [0, 1] to a space X. α : I → X, where α(0) = α(1) = x₀. Two loops α and β are homotopic if a continuous map between them exists. This homotopic equivalence can be extended to arbitrary maps: f, g : X → Y . f and g are homotopic if there exists a continuous map between them. The fundamental group or first-homotopy group of the space X at x₀ is denoted by π₁(X, x₀). This group comprises all homotopic equivalent mappings and is characterized by a space isomorphic to it. We can drop the x0 label if the group X is arcwise connected, which in our case will be true in most cases.¹

Let us start with a simple example: the space X = S¹, a 1D sphere, i.e. a circumference, is isomorphic to Z: π1(S¹) ∼= Z. All loops can be described by an integer counting the number of times that the loop covers the entire circumference. Clockwise or counter-clockwise direction change the sign of the integer. This invariant is called the winding number, it counts how many times the loop winds around the origin. See Fig. 2.1.

1Only for zero dimensional systems will we consider arcwise disconnected spaces as we will explain later.

23

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24 CHAPTER 2. FUNDAMENTALS OF TOPOLOGY AND MECHANICAL METAMATERIALS

Figure 2.1: Circumference space, S¹, with several loops starting at colored dots x₀. The one starting in the blue dot, winds one time counter-clockwise, its winding invariant is +1. The loop starting in the green dot, winds two times clockwise, its winding number is −2. The loop starting in the orange dot, does not wind around the center, its winding invariant is 0.

Figure 2.2: Torus space, T², with several loops and winding numbers. The topological invariant consist of two integers representing two possible loops that are not isomorphic to each other. These two nonequivalent loops are indicated by dashed line, with topological invariant (0, 1), and dot-dashed line with topological invariant (1, 0). The loop with double-dot-dashed line is reducible to a point and its topological invariant is (0, 0).

Now, if our space X is a sphere S², its fundamental group π₁(S²) ∼= 0 is trivial because all loops in a sphere can be contracted to a point. In contrast, if X is a donut (i.e. a torus X = T²), π1(T²) ∼= Z ⊕ Z in the same way as if X were a cup, see Fig. 2.2.

Higher homotopy groups study n-loop structures, defined as a mapping α : I ⊗ I ⊗ . . . ⊗ I → X.

Where the starting space is the product of n I-segments. Two n-loops are homotopic if there exists a continuous mapping between them. In general, the target space of spheres fulfills π_n(Sⁿ) ∼= Z.

Homotopy theory has been employed in condensed matter theory to classify elements of the set of projectors within a given symmetry class of Hamiltonians; i.e. whether a given element (Hamiltonian) can be continuously deformed into any other without closing the energy gap or not. Homotopy theory also classifies defects such as solitons, vortices or monopoles in condensed systems. These two cases are explained in the following two sections.

This section is based on the book [1].

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2.2. TOPOLOGICAL DEFECTS 25

Figure 2.3: Different configurations of x − y model a) state of total alignment isomorphic to any configuration without singularities b) vortex distortion, the center is represented by a red circle.

c) domain wall of Ising type.

2.2 Topological defects

A classic example of a topological defect is a vortex in the x − y model [1, 2]. The x − y model is comprised of two dimensional arrows defined in a plane s(~r), where s is a two component vector of length 1. A single parameter φ(~r) defines the state of the system, s(~r) = (cos(φ(~r), sin(φ(~r))), see Fig. 2.3. We can define φ(~r) = nθ + φ0 where θ is the angular coordinate in polar representation.

The origin of coordinates does not have a well defined θ and a core is needed where the parameter s vanishes. For instance, s(~r) = s(r, θ) = tanh(^r_ξ)(cos(φ(~r)), sin(φ(~r))) where ξ is the size of the core.

Then, the phase changes by 2πn when going around a circle enclosing the center, see Fig. 2.3 b), I

C

dφ = 2πn. (2.1)

This defect cannot be made to disappear by any continuous deformation. It is not possible to recover the n = 0 state from the n = 1 vortex without flipping the arrows discontinuously. Let us see that it is a topological defect.

We apply homotopy theory to this case. The initial space is I = φ = 2π[0, 1]. The target space is the space of the order parameter s(~r), which is isomorphic to a circle, i.e. X = S¹. Therefore, the homotopy group we are interested in is π₁(S¹) ∼= Z characterized by integers. Consequently, mathematical topology guarantees that a vortex n cannot be connected continuously with other integer value.

In general, an m-dimensional defect embedded in a d-dimensional space is classified by a π_n(X), where n = d−m−1 [1]. For the x−y model, the defect or the core is zero dimensional m = 0 and the medium is bidimensional d = 2. Other cases would be line vortices m = 1 in three dimensional superconductors d = 3 or domain walls m = 1 (m = 2) in d = 2 (d = 3) systems. Of particular interest in this thesis are the cases of vortices and domain walls in two dimensional systems. The former is characterized by π1(S¹) ∼= Z and the latter by π0(X). The meaning of zeroth-homotopy is slightly different [1], it maps from I⁰ = {0} a point to the space X. Two zeroth-order loops are α, β : I⁰ → X such that α(0) = x0 and β(0) = x1. A continuous mapping between the two zeroth-order loops exists if and only if x0 and x1 are arcwise connected in the target space X. This way, π0(X) ∼= 0 implies that X is arcwise connected, π₀(X) ∼= Zn implies that there are n + 1 different arcwise connected components.

For instance, in the Ising model (a particular case of x − y model) the degrees of freedom can only take two possible values X = {−1, +1} = S⁰ and π0(S⁰) ∼= Z2. See Fig. 2.3 c).

2.3 Electronic topology

A new kind of quantum materials became relevant with the understanding of the integer quantum Hall effect, the topological insulators. These topological phases have properties to which an invari-

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26 CHAPTER 2. FUNDAMENTALS OF TOPOLOGY AND MECHANICAL METAMATERIALS ant can be assigned. This invariant depends only on global properties and it is robust against local perturbations such as disorder and scattering as long as the bulk gap is not closed. They can have surface metallic states, which distinguishes them from ordinary insulating phases.

The first example of a topological phenomenon is the integer quantum Hall effect [3, 4, 5]. Elec- trons moving in two dimensions with a strong perpendicular magnetic field showed quantization of the transverse conductance with remarkable accuracy. This effect has a topological explanation. A electronic topological invariant, computed from band eigenvectors over the Brillouin Zone, had phys- ically observable consequences. It revealed that the phase of eigenvalues, considered unimportant for physical properties until then, are relevant for topological phenomena.

A Hamiltonian describing the quantum Hall effect does not have any required symmetry, apart from translation symmetry. In the presence of translation invariance, ground states of free fermionic systems can be seen as a filled Fermi sea in the d-dimensional Brillouin zone (BZ), in Fourier space. The band structure can be understood as a map from the BZ to the space of Bloch Hamiltonians: f : k^d→ H.

Notice that a d dimensional BZ fulfills the periodicity requirements of d-loops for d-th homotopy group.

The spectral projection operator, Q, which is defined as

Q(k) = X

k,j=filled

|u^j_ki hu^j_k| − X

k,j=empty

|u^j_ki hu^j_k| , (2.2)

contains the topological information but the bands are flattened to ±1. Similarly, a map from the reciprocal unit cell to the space of projectors or target space [6, 7] g : k^d→ Q is characterized by the same homotopy group. Both maps, f and g, contain equivalent topological information.

The projector of a Hamiltonian with m occupied levels and n empty levels is a unitary matrix U (m+n).

We still need to consider the gauge symmetry existing between occupied levels and similarly for empty levels. Thus, each projector is an element of the coset U (m + n)/U (m) × U (n).² For two spatial dimensions (where the IQHE occurs) the relevant homotopy group is π₂(U (m + n)/U (m) × U (n)) ∼= Z, which is isomorphic to the integer set. The projectors are topologically classified by an integer or Chern number. Projectors (or Hamiltonians) with different Chern numbers cannot be deformed into each other continuously. This is the mathematical reason why there exists an IQHE in 2D, and the Hall con- ductivity σ_xy is an integer in natural units of e²/h. On the contrary, π₁(U (m + n)/U (m) × U (n)) ∼= 0 and π3(U (m + n)/U (m) × U (n)) ∼= 0 are trivial, meaning there is no such effect in 1D or 3D.

To study other symmetry classes, we impose restrictions on the form of Q(k) operator, determine the coset where all projectors with those symmetries are included and study the homotopy group relevant for each dimension. Unitary symmetries only reduce the dimension of the Hamiltonian, but time reversal symmetry (TRS), particle–hole symmetry (PHS) and sublattice symmetry (SLS)³ act differently. They impose constraints, forcing the Hamiltonian to be real and block off-diagonal. The combination of these three symmetries results in a tenfold classification. This work was carried out by the French mathematician Cartan in 1926 and the notation is still used nowadays. See table 2.1.

TRS and PHS are anti-unitary operators that can square to +1 or −1, the former commutes with the Hamiltonian, the latter anticommutes. SLS is the product of the two, with the exception that when the two are absent, SLS can exist, which is the AIII class. TRS squaring to = +1 indicates that the SU(2) spin is an integer and TRS squaring −1 indicates the Hamiltonian degrees of freedom have half- integer spin. The Bogoliubov—de Gennes type of Hamiltonians represent superconductors and have non-zero PHS; they belong to classes C, CI, D and DIII. These Hamiltonians preserve SU(2) spin-1/2 rotation symmetry when PHS is -1, while it does not preserve SU(2) when PHS equals +1 [8, 6, 9].

2This coset is isomorphic to Grassmannian Gm,m+n(C) which is commonly used in the literature.

3Also called chiral symmetry

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2.3. ELECTRONIC TOPOLOGY 27 Table 2.1: Classification table of topological insulators and superconductors as a function of spatial dimension d and symmetry class (first column). The definition of the ten generic symmetry classes of single particle Hamiltonians according to their behavior under time-reversal symmetry (TRS), charge- conjugation or particle—hole symmetry (PHS), as well as sublattice or chiral symmetry (SLS). Two groups are distinguished, the complex and real cases. The symmetry classes are ordered in such a way that a periodic pattern in dimensionality is visible.

TRS PHS SLS d=0 d=1 d=2 d=3 d=4 d=5 d=6 d=7 d=8

A 0 0 0 Z − Z − Z − Z − Z

AIII 0 0 1 − Z − Z − Z − Z −

AI +1 0 0 Z − − − 2Z − Z2 Z2 Z

BDI +1 +1 1 Z2 Z − − − 2Z − Z2 Z2

D 0 +1 0 Z2 Z2 Z − − − 2Z − Z2

DIII −1 +1 1 − Z2 Z2 Z − − − 2Z −

AII −1 0 0 2Z − Z2 Z2 Z − − − 2Z

CII −1 −1 1 − 2Z − Z2 Z2 Z − − −

C 0 −1 0 − − 2Z − Z2 Z2 Z − −

CI +1 −1 1 − − − 2Z − Z2 Z2 Z −

The table has periodic behavior, complex classes (A and AIII) are those without TRS or PHS. They have period two in the dimension of the system. The topological invariant of class A in dimension d is the same as class AIII in d + 1 and A in d + 2. For real classes, the period is eight. The topological invariant in a real class in dimension d is identical with dimension d + 8. (Observe the d = 8 column is identical with d = 0 column in table 2.1). Furthermore, there are dimensional reduction arguments to understand, with the same invariant, other topological class in dimension d − 1. Notice that the table emphasizes the periodic behaviour responsible for invariants appearing in diagonal lines. In some cases, it is possible to derive the topological invariant in d − 1 dimensions conserving all symmetries and, therefore, the topological class [7]. This latter case will be relevant in the BDI case and d = 1.

2.3.1 The BDI class

In topological mechanics, we will study particular systems that are isostatic. A particular mathematical construction allows to study these systems as BDI class Hamiltonians [8, 6, 9], as we will explain later. BDI class is characterized by real in real space and block off-diagonal Hamiltonians. In the classification table, table 2.1, TRS and PHS square to +1 and sublattice symmetry is present. Spinless SSH model is a model belonging to this class.

The SSH model

This model of polyacetylene, proposed by Su, Schrieffer and Heeger [10, 11], describes spinless fermions hopping on a 1D lattice with staggered hopping amplitudes, see Fig. 2.4. In a model with N unit cells and 2 sites per unit cell (UC), A and B, the Hamiltonian reads,

H =

N

X

j=1

tc^†_A,jcB,j+ t⁰c^†_B,jcA,j+1+ h.c. − µ

c^†_A,jcA,j + c^†_B,jcB,j

. (2.3)

We fix the number of fermions to half-filling N , by setting the chemical potential µ to zero. In the periodic infinite system and momentum space the Hamiltonian reads,

H =X

k

c^†_A,k, c^†_B,k

0 t + t⁰e^ika t + t⁰e^−ika 0

c_A,k cB,k

. (2.4)

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28 CHAPTER 2. FUNDAMENTALS OF TOPOLOGY AND MECHANICAL METAMATERIALS The Hamiltonian is block off-diagonal due to sublattice symmetry. This symmetry is reflected in the projector operator Q(k) that is block off-diagonal as well when sublattice symmetry is present. We call the upper off-diagonal part of Q(k), q(k). As long as t 6= t⁰, there is a gap in the spectrum, and the system is an insulator. The band structure is E±(k) = ±pt²+ t⁰²+ 2tt⁰cos(ka) and the gap size is ∆ = 2|t − t⁰|. Although the band structure is useful to know many physical properties, it does not contain all the information.

The topological information can be seen in different ways,

1. The Hamiltonian is 2 × 2, it can be represented in the basis of Pauli matrices plus the identity matrix, H = P

jhjσj, where j = {0, 1, 2, 3}. The identity matrix, σ0 plays the role of the chemical potential and h0 can be fixed to zero. The other diagonal term h3σ3 must be zero⁴ in order for the system to be sublattice symmetric. This way, there are only two independent parameters defining uniquely the Hamiltonian. The parameter space is R²or C, i.e. the mapping goes k → R² : (h1(k), h2(k)) or k → C : h1(k) + ih2(k) both paths form a closed loop due to the periodicity of k in the BZ. Moreover, the system is gapped as long as h₁ 6= 0 6= h₂, and since the actual value of the parameters is irrelevant, there is a unique way of projecting the loop in C with the unitary circumference. We can say that the space of Hamiltonians describing SSH insulating phase is isomorphic to S¹. k → S¹ : e^iθ(k), where θ(k) is the angular polar coordinate of (h₁, h₂). Hence, the fundamental homotopy group of this Hamiltonian space is π1(S¹) ∼= Z and it is classified by integer invariants, as we knew from the classification table.

We can compute the invariant of a particular model by seeing how many times the loop winds around the origin.

ν = 1 2πai

Z π/a

−π/a

dk∂_klog h(k), (2.5)

where h(k) = h1(k) + ih2(k) = (t + t⁰cos(ka)) + i(t⁰sin(ka)).

2. The eigenvectors in Hilbert space carry the topological features. The eigenvectors are

u^±_k = 1

√2

±1

t+t⁰e^−ika

|E|

!

. (2.6)

The so-called Zak phase or Berry phase in 1D is defined as γj = 2πi

Z π/a

−π/a

hu^j_k|∂_k|u^j_ki dk (2.7)

for each band j = {+, −}. The topological invariant for the Hamiltonian is ν = _2πi^γ⁻ proportional to the sum of Zak phases of all occupied bands (there is an integer number of them for insulators).

This method is general, not restricted to 2 × 2 Hamiltonians in the BDI class.

3. Let us consider a path of q(k) as k goes through the Brillouin Zone. Notice that due to the periodicity of the BZ, this mapping defines a loop in the target space.

Q(k) = |u⁻_ki hu⁻_k| − |u⁺_ki hu⁺_k| = 0 −^t+t⁰_|E|^e^−ika

−^t+t⁰_|E|^e^−ika 0

!

, (2.8)

where q(k) = −^t+t⁰_|E|^e^−ika. We will ignore the modulation |E| which is always positive and can be seen as a change in scale that does not change the topology. The path q(k) is then a loop of

4It could be non-zero if any of the others components, h1, h2 are zero. Nonetheless, a rotation in the σ space must exists in which h3 is zero.

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2.3. ELECTRONIC TOPOLOGY 29

Figure 2.4: Schematic representation of the SSH model.

radius t⁰ in the complex plane, with center at t. This loop lies on one plane, due to sublattice symmetry. We have considered the hopping parameters, t and t⁰ real, but the argument is general. The center is enclosed by the loop if |t⁰| > |t| and Hamiltonians are not continuously connected with those not enclosing the center |t⁰| < |t|. A loop mapping to the center means that the Hamiltonian is gapless, another option is the loop moving out of the plane, meaning breaking sublattice symmetry including a non-zero h₃ component.

This intuitive argument distinguishes two different topological phases of SSH model (as in table 2.1). If we look at the periodic classification table in BDI row, we see a non-trivial integer invariant in 1D. The topological invariant reads,

ν = Z

BZ

q(k)⁻¹∂_kq(k)dk, (2.9)

ν is the winding number, because it gives the number of times the loop mapping from BZ to q(k) space winds around the origin. In 1D, only the fundamental homotopy group is relevant.

We have ν = 1 in the non-trivial phase |t⁰| > |t| and ν = 0 in the trivial phase |t⁰| < |t|.

The three methods are equivalent and can be generalized to higher dimensional Hamiltonians ν = 1

2πai Z π/a

−π/a

dk det [∂_klog h(k)] = Z

BZ

Trq(k)⁻¹∂_kq(k) dk. (2.10) One important feature of topology is the bulk-edge correspondence. When we compute non-trivial invariants in the bulk of a Hamiltonian, the open system will have zero energy edge states. In this BDI model, the edges are zero dimensional and the edge states are localized modes at the edge. Let us consider the dimerized case in the non-trivial phase: |t⁰| |t|. In the limiting case where t = 0, an orbital state is isolated at each side of the open system, see Fig. 2.5. These isolated states have zero energy or mid-gap energy. That state is localized at the edge because the bulk bands are energetically separated E±= ±|t⁰| and the overlap is nonexistent. If we turn on the intra-cell hopping t, the bands will disperse, and as long as the gap is open the overlap will be exponentially decaying. An insulator is needed to protect the topological states.

Higher dimensional lattices have more complicated topological invariants than the SSH model, corre- sponding to higher order homotopy mappings.

2.3.2 Chern insulator and magnetic monopoles

The purpose of this section is to introduce the Berry curvature which will be important in mechanical systems.

As we mentioned before, a Chern insulator belongs to class A. Hamiltonians in this class do not have any symmetry beyond translation invariance. In two dimensions it has an integer topological invariant.

A particular and illustrative example is a 2 × 2 Hamiltonian. It could be written in terms of Pauli matrices.

H =X

j

h_jσ_j. (2.11)

The minimal model, disregarding the constant term, is a 3 parameter Hamiltonian R³, i.e. j = {1, 2, 3}.

Its spectrum is E = ±ph²₁+ h²₂+ h²₃ and it is gapped as long as the three components do not vanish.

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30 CHAPTER 2. FUNDAMENTALS OF TOPOLOGY AND MECHANICAL METAMATERIALS

Figure 2.5: a) Energy states of the periodic SSH chain for varying the hopping parameters ratio t⁰/t, the gap closes and reopens with two midgap states. The gap closing indicates a topological phase transition and zero-energy states appear for non-zero winding number. b–c) a real space representation of the SSH chain dimerization in the limiting cases with parameters b) t⁰ = 0 c) t = 0. In the latter case two zero energy states emerge at the edges.

The absolute value is irrelevant for topology (it can be deformed continuously), thus the projection in the unitary sphere is the target space. The surface loop from k² to S² is described by the second homotopy group and π₂(S²) ∼= Z.

Higher dimensional Hamiltonians can be described by groups of anticommuting matrices with similar properties to Pauli matrices. These are Γ-matrices and the deduction of the kind of invariant is equivalent [1, 6, 7].

A minimal model with non trivial topology is described by the following parameters, h1(~k) = ∆ sin(kxa),

h2(~k) = ∆ sin(kya),

h₃(~k) = −2t cos(k_xa) − 2t cos(k_ya) − µ,

(2.12)

with t and ∆ real and positive and µ real. Notice that h_j(k_x, k_y) are periodic functions in the square Brillouin Zone, see Fig. 2.6. The eigenvector for the lower energy band reads,

u⁻

I~k= 1 N_I

h3(~k) − |E(~k)|

h1(~k) + ih2(~k)

!

. (2.13)

This eigenvector is singular if h₁ = h₂ = ^π_a and h₃ > 0, which is the case at k_x = k_y = 0 and µ < 4t.

We can multiply the eigenvector by a phase and the physics remains unchanged. An appropriate phase will solve the singularity problem at k_x = k_y = 0.

u⁻

II~k= 1 N_II

−h₁(~k) + ih2(~k) h₃(~k) + |E(~k)|

!

. (2.14)

However, this new eigenvector is singular at kx = ky = 0 and −4t < µ. This singularity is unavoidable but it is not physical either. The phase is not directly observed and the eigenvector can be defined by parts in the Brillouin Zone, see Fig. 2.6,

u⁻_~

k = (u⁻

I~k if |~k| < k0, u⁻

II~k otherwise , (2.15)

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2.3. ELECTRONIC TOPOLOGY 31 where k₀ defines a circle in the BZ such that k₀a < π√

2, i.e. the circle does not enclose the singularity point in u⁻_I. This singularity problem is not a coincidence, there is a region in parameter space

−4t < µ < 4t that has unavoidable singularities in which they need to be defined by parts.

Let us rewrite the eigenvectors as Bloch waves, ψ(~r) =X

~k

u_~_k(~r)e^−i~^k·~^r (2.16)

to emphasize the connection with the real space wave function ψ(~r) and the decomposition of the Hamitonian in ~k-space.

The same derivation would follow from the eigenvectors of the spectral projector Q(~k). As can be seen from its definition, Eq. (2.2), it has identical eigenvectors.

To introduce the physical properties we can measure or compute, let us see where a non-physical singularity could appear in classical theory. More precisely, let us imagine a magnetic monopole and its equations. If we compute the flux of the magnetic field though a closed surface S,

I

S

B · d ~~ S = Z

V

∇ · ~BdV = 0, (2.17)

and make use of the divergence theorem, we end up with the volume integral of the divergence of the magnetic field. We learned that the divergence of the magnetic field ∇ · ~B is zero (second Maxwell equation), because there are not magnetic monopoles. Let us assume there are. First,

q_m = I

S

B · d ~~ S (2.18)

implies that the magnetic field is

B(~~ r) = qm

4π²

~ er

r², (2.19)

where qm is a magnetic “charge”, ~er is the radial unitary vector in spherical coordinates and r is the radial coordinate. For the dynamics it is important the vector potential, ~B = ∇ × ~A. ~A is defined up to a gauge transformation ~A⁰ = ~A + ∇Λ where Λ is any scalar function in real space.

A~_I = q_my~e_x− x~e_y

r(r − z) (2.20)

is a solution that meets the requirements of the expected magnetic field. Nevertheless, it is singular at r = z. If we apply a gauge transformation, then we can define

A~_II = q_m−y~e_x+ x~e_y

r(r + z) , (2.21)

which is singular at z = −r. Yet, this is not a problem because the singularity lies in the vector potential, which is not an observable. We can define the vector potential by parts to avoid singularities and compute the physical magnetic field. Furthermore, due to topological reasons the integral of the magnetic field in a closed surface is quantized (analogous to Gauss-Bonnet theorem or Gauss law).

Going back to our Chern insulator, we can elucidate several similarities. First, the existence of singularities in non-observable quantities: the vector potential and the eigenvectors. Second, the definition by parts connected by gauge transformations. Actually, we can find an expression for an analogue to the vector potential that fulfills the gauge invariance of the eigenvectors,

u_~⁰

k= u_~_ke^iφ(~^k). (2.22)

Topological properties of mechanical metamaterials

NATALIA LERA VALVERDE

TESIS DOCTORAL MADRID 2019

TOPOL OGICAL PR OPER TI ES OF ME CH ANICAL MET AMA TERIALS NA TALIA LE RA

TOPOLOGICAL PROPERTIES OF MECHANICAL METAMATERIALS

UNIVERSIDAD AUTÓNOMA DE MADRID FACULTAD DE CIENCIAS

Topological properties of mechanical metamaterials Natalia Lera

Topological mechanics is a recent and intense area of research. It merges the knowledge of mathematical topology, electronic topology and topological defects with mechanical systems and phononic band structures.

In this thesis, we strive to understand the basic ingredients for

non-trivial topology in mechanical systems arising from both

topological defects and band theory. As a result, new mechanical

models have been proposed and analyzed theoretically.

Topological properties of mechanical metamaterials

Contents

Acknowledgments

List of Figures

List of symbols and abbreviations

Chapter 1

Introduction

Introducci´ on

Chapter 2

Fundamentals of Topology and Mechanical Metamaterials

2.1 Mathematical concepts

2.2 Topological defects

2.3 Electronic topology